The Cauchy problem for the generalized Ostrovsky equation with negative dispersion

TL;DR

This paper investigates the well-posedness and solution behavior of the generalized Ostrovsky equation with negative dispersion, establishing local well-posedness in specific Sobolev spaces and convergence to the generalized KdV equation as the rotation parameter diminishes.

Contribution

It proves local well-posedness in Sobolev spaces and introduces a new function space for the analysis, also demonstrating the solution's convergence to the generalized KdV equation as the rotation parameter approaches zero.

Findings

Local well-posedness in H^s for s > 1/2 - 2/k

Well-posedness in the space X_s incorporating Fourier transform properties

Solution convergence to generalized KdV as gamma tends to zero

Abstract

This paper is devoted to studying the Cauchy problem for the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta\gamma<0,\gamma>0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^{s}(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_{s}(\mathbb{R}): =\|f\|_{H^{s}}+\left\|\mathscr{F}_{x}^{-1}\left(\frac{\mathscr{F}_{x} f(\xi)}{\xi}\right)\right\|_{H^{s}}\left(s>\frac{1}{2}-\frac{2}{k}\right).$ Finally, we show that the solution to the Cauchy problem for generalized Ostrovsky equation converges to the solution to the generalized KdV equation as the rotation parameter $\gamma$ tends to zero for data belonging to $X_{s}(\mathbb{R})(s>\frac{3}{2})$. The main difficulty is that the phase function of Ostrosvky equation with negative dispersive $\beta\xi^{3}+\frac{\gamma}{\xi}$ possesses the zero singular point.